Paradoxical Game Show Problem Here's a problem that has had me scratching my head for a long time:
Imagine you're in a game show, and are presented with 2 boxes.  You are told that both boxes contain a sum of cash, but one of the boxes contains twice as much as the other.  You do not know which box has the double prize.  The game works in 2 phases:


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*Choose any of the boxes you want.

*Look inside the box.  At this point you can decide to keep the contents, or switch to the other box.


So imagine that you've chosen a box, and it contains \$100.  From here, you can calculate the "expected value" of the other box to be $0.5 \times \$50 + 0.5 \times \$200 = \$125$ and therefore decide to switch.
But then it follows that you would have made the same decision for any value $x$ that you would have found in the first box!  So then why not just pick the other box in the first place?
In other words, the strategy of "pick a box at random, and then switch, no matter what" is equivalent to "pick a candidate box at random, and then pick the other box, and keep it", which is also equivalent to "pick a box at random, and keep it".  Which means that switching is the same as not switching.
But this seems like a paradox, because we just calculated that switching the box after your initial choice increases your expected winnings by a factor of 1.25!
 A: Here is how you can view this in order to make it right for you:


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*The total amount of money in both boxes is $3X$ dollars

*One box contains $X$ dollars and the other box contains $2X$ dollars

*Now you pick a box, and you're thinking "maybe I should pick the other box":


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*There is a $50$% chance that the other box contains $\frac{1}{3}$ of the amount

*There is a $50$% chance that the other box contains $\frac{2}{3}$ of the amount


*So the expected amount of money that you'll get by picking the other box is $\frac{1}{2}\cdot\frac{1}{3}+\frac{1}{2}\cdot\frac{2}{3}=\frac{1}{2}$

Note that $\frac{1}{2}$ of the amount of money is the expected (average) portion that you win.
In essence, you will win either $\frac{1}{3}$ of the amount of money or $\frac{2}{3}$ of the amount of money.
A: This simply because average in terms of probabilty doesn't work. Like if you use an averrage by getting the square root of the product of 200 and 50 you get 31.6... Using average in probabilty only works if you do it like 'barak manos' (right below this post).
